Once you have determined your students’ problem-solving difficulties, the next step is to determine these difficulties have been addressed in some existing research-based, problem-solving frameworks. There are several research-based frameworks
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Part 4: Personalize a Problem Solving Framework and Problems Figure 14.5. Three examples of research-based problem-solving frameworks.1. UNDERSTAND
for physics that have been developed and successfully used,iv,v,vi all based on the general framework developed by George Polya,vii Two examples are outlined in Figure 14.5. Each framework divides the important actions into a different number of steps and sub-steps, describes the same actions in different ways, and emphasizes different heuristics depending on the backgrounds and needs of population of students for whom they were developed.
For each framework consider:
the specific actions in each step;
how these actions are carried out in solving a physics problem;
the specific problem-solving difficulties each step and/or action is designed to help students overcome;
the match with the problem-solving difficulties your students exhibit.
Step . Construct Your Problem-solving Framework
Begin with a Research-based Framework. Choose the research-based, problem-solving framework that best addresses the major difficulties of the majority of your students.
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Figure 14.6. Outline of the Competent Problem-solving Framework: Calculus Version
1. Focus the Problem. Establish a clear mental image of the problem.
A. Visualize the situation and events by sketching a useful picture.
• Show how the objects are related spatially, Show the time sequence of events, especially the initial and final states of the objects. Add times to the drawing when an object experiences an abrupt change in interaction.
• Write down the known information, giving each quantity a symbolic name and adding that information to the picture.
B. Precisely state the question to be answered in terms you can calculate.
C. Identify physics approach(es) that might be useful to reach a solution.
• Which fundamental principle(s) of physics (e.g., kinematics, Newton’s Laws, conservation of energy) might be useful this problem situation.
• List any approximations (e.g., assume kinetic friction is negligible) or problem constraints (e.g., constant acceleration, T1 = T2, uniform electric field) that are reasonable in this situation.
2. Describe the Physics
A. Draw any necessary diagrams (e.g., motion diagram, force diagram, momentum diagram, energy table) with coordinate systems that are consistent with your approach(es).
• Define consistent and unique symbols for any quantities that are relevant to the situation.
• Identify which of these quantities is known and which is unknown.
B. Identify the target quantity(s) that will provide the answer to the question.
C. Assemble the appropriate equations that mathematically give the physics principles, approximations, and problem constraints identified in your approach.
3. Plan a Solution
A. Construct a logical chain of equations from those identified in the previous step, leading from the target quantity to quantities that are known.
• Choose an equation that contains the target quantity and write it down. Identify other unknowns in that equation.
• Choose a new equation for one of these unknowns. Write down this equation and note the unknown quantity this equation was chosen to determine.
• Continue this process for each unknown.
B. Determine if this chain of equations is sufficient to solve for the target quantity by comparing the number of unknown quantities to the number of equations.
4. Execute the Plan
A. Follow the outline from in the previous step.
• Arrive at an algebraic equation for your target quantity by following your chain of equations in reverse from the order you constructed your plan.
• Check the units of your algebraic equation before putting in numbers.
• Use numerical values to calculate the target quantity.
5. Evaluate the Answer
A. Does the mathematical result answer the question asked with appropriate units?
B. Is the result unreasonable?
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Part 4: Personalize a Problem Solving Framework and Problems Figure 14.7. The problem-solving framework by Fred Reiffor a calculus-based introductory course in mechanics.
1. Analyze the Problem: Bring the problem into a form facilitating its subsequent solution.
A. Basic Description—clearly specify the problem by
• describing the situation, summarizing by drawing diagram(s) accompanied by some words, and by introducing useful symbols; and
• specifying compactly the goal(s) of the problem (wanted unknowns, symbolically or numerically)
B. Refined Description—analyze the problem further by
• specifying the time-sequence of events (e.g., by visualizing the motion of objects as they might be observed in successive movie frames, and identifying the time intervals where the description of the situation is distinctly different (e.g., where acceleration of object is different); and
• describing the situation in terms of important physics concepts (e.g., by specifying information about velocity, acceleration, forces, etc.).
2. Construct a Solution: Solve simpler sub-problems repeatedly until the original problem has been solved.
A. Choose sub-problems by:
• examining the status of the problem at any stage by identifying the available known and unknown information, and the obstacles hindering a solution;
• identifying available options for sub-problems that can help overcome the obstacles;
• selecting a useful sub-problem among these options.
B. If the obstacle is lack of useful information, then apply a basic relation (from general physics knowledge, such as Fnet = ma, fk = N, x = 1/2 axt2) to some object or system at some time (or between some times) along some direction.
C. When an available useful relation contains an unwanted unknown, eliminate the unwanted quantity by combining two (or more) relations containing this quantity.
Note: Keep track of wanted unknowns (underlined twice) and unwanted unknowns (underlined once).
3. Check and Revise: A solution is rarely free of errors and should be regarded as provisional until checked and appropriately revised.
A. Goals Attained? Has all wanted information been found?
B. Well-specified? Are answers expressed in terms of known quantities? Are units specified?
Are both magnitudes and directions of vectors specified?
C. Self-consistent? Are units in equations consistent? Are signs (or directions) on both sides of an equation consistent?
D. Consistent with other known information? Are values sensible (e.g., consistent with known magnitudes)? Are answers consistent with special cases (e.g., with extreme or specially simple cases)? Are answers consistent with known dependence (e.g., with knowledge of how quantities increase or decrease)?
E. Optimal? Are answers and solution as clear and simple as possible? Is answer a general algebraic expression rather than a mere number?
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Language. Next consider how your students would interpret language calling for the specific actions in the framework. Since the framework was developed as a specific implementation of Polya’s general framework used by experts in all fields, it will probably need to be modified for your particular population of students.
1. Retain the elements of each research-based framework that:
• seem directed at your students’ most common problem-solving difficulties; and
• are consistent with your course goals and approach.
2. Reduce or eliminate steps in the framework that address difficulties not exhibited by your students.
3. Make sure that the framework describes a complete, logical problem-solving procedure from your point of view. Fill in steps if necessary to assure completeness.
4. Choose language to describe the steps and actions of your modified problem-solving framework that is most meaningful to you and your students (see also Step ).
Compare with Research-Based Frameworks. Make sure that all of the features of a research-based framework are incorporated (see Chapter 4, pages //-//).
Check Your Framework. Check that the framework will be useful in all parts of your course by solving problems using its procedure for different physics topics.
Remember, a problem-solving framework is most useful at the beginning of the course. When students become more familiar with the framework and comfortable using the framework, the different steps begin to merge. For example, towards the end of the first semester, our students in the calculus-based course merge Steps 1 and 2 into one description, and merge Steps 3 and 4 into one procedure. This is an example of our 2nd Law of Instruction: Don’t change course in midstream; structure early then gradually reduce the structure
Example 1. Compare Algebra versus Calculus Versions
Figure 14.6 outlines our research-based Competent Problem-solving Framework for students in our calculus-based course.viii Compare this framework with the outline of our framework for students in our algebra-based course (Chapter 4, page //).ix Steps 1, 2, 4, and 5 in these frameworks are identical because we found that students in both our algebra-based and calculus-based courses had similar conceptual
difficulties. These difficulties, addressed in steps 1, 2, and 5, were in visualizing a physical situation, connecting physics to reality, recognizing a common physics theme, applying fundamental concepts, and integrating knowledge into a coherent conceptual framework.
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Part 4: Personalize a Problem Solving Framework and ProblemsIn addition almost all students had difficulty solving problems in a logical, organized progression. These difficulties were addressed in steps 3 and 4 of the framework.
The actions in step 3 are slightly different because the calculus-based students have more dexterity in mathematics than the algebra-based students. Most of the
calculus-based students (but not all) could follow their plan without needing to write an outline of their actions. The algebra-based students, on the other hand, needed this organizational tool to keep from becoming consumed by superfluous
mathematical manipulation.
Example 2. Compare Two Calculus-based Versions
Figure 14.7 contains an outline of another research-based framework developed by Fred Reif and his group at Carnegie Mellon University3 for use in the first semester (mechanics) calculus-based course. In contrast, our Competent framework (Figure 14.6) is generalized for use in both semesters of an introductory course. Reif’s framework breaks the generalized procedure into three major steps, whereas our Competent framework has five major steps (see Figure 14.5).
Fred Reif’s Framework Our Competent Framework
One major difference between the two frameworks is in the heuristic emphasized to plan and construct a solution. A heuristic is a rule of thumb – a procedure that is both powerful and general, but not absolutely guaranteed to work. (see Chapter 4, pages // - //). Reif’s framework emphasizes the heuristic of breaking a problem into sub-problems that you can solve (Step 2). Our Competent framework emphasizes the heuristic of working backwardsx from the goal to the solution (Step 3). Working backwards is a powerful heuristic for students who have do not know where to start their mathematics solutions when faced with a problem that is slightly different than the example problems and solutions they encounter in class. Apparently the
freshman students at Carnegie Mellon University do not use the pattern-matching novice strategy as much as our freshmen.
Another difference in the two frameworks is how they distinguish physics principles that apply in many topics (e.g., kinematics, Newton’s second Law, conservation of energy) versus relationships that apply only in specific problem situations (e.g., constant force, potential energy when the gravitational interaction is near the Earth’s surface). In Reif’s framework, this difference is not emphasized, whereas in our Competent framework the difference is emphasized. Consequently, there are two steps in the Competent framework (illustrated in Steps 1C and 2C) and only one
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step in Reif’s framework (illustrated in Steps 1B and 2B). There are two reasons for this difference.
1. Reif’s framework is limited to one semester of mechanics.
2. The prominence of principles is part of our overarching goal of teaching physics through problem solving. Many students enter our courses believing that physics is a collection of disconnected concepts and equations. One of our goals is to help dispel this belief by approaching all topics with the same fundamental physics principles. [Here we go again. Lets see how the conservation of energy applies to electric circuits.]
Despite the differences in the two frameworks, the contents of the frameworks are remarkably similar. The two frameworks were developed independently at about the same time, as was the framework by Alan Van Heuvelen.1 These problem-solving frameworks are based on the same problem-solving research, consequently they are similar in content, if not in specific heuristics and language.